Note on vanishing power sums of roots of unity

Abstract

For a given positive integers $m$ and $\ell$, we give a complete list of positive integers $n$ for which their exist $m$th roots of unity $x_1,\dots ,x_n\in \mathbb{C}$ such that $x_1^{\ell }+\cdots +x_n^{\ell }=0$. This extends the earlier result of Lam and Leung on vanishing sums of roots of unity. Furthermore, we prove that for which integers $n$ with $2\le n\le m$, there are distinct $m$th roots of unity $x_1,\dots ,x_n\in \mathbb{C}$ such that $x_1^{\ell }+\cdots +x_n^{\ell }=0$.

arXiv comment5 pages, 1 table, to appear in Math. Student

References in corpus (1)

• Prime ideals and regular sequences of symmetric polynomials